introduction to differential equations

EMDADUL HAQUE MILON
ehmilon24171@gmail.com
mdehmilon24171@gmail.com
DEPERTMENT OF STATISTICS,
UNIVERSITY OF RAJSHAHI..
RAJSHAHI, BANGLADESH
6205&6206
The members of Group-A
• Kaderi kibria Farjana Yasmin moni Md. Shadikug jaman Mst. Afia ibnat
• Mst. Sumona sumi Minhazul islam Md. Ayub ali nayan Md. Dinar
Introduction to Differential
Equations
•Topics:
•Basic concept of differential equation,
•Classification,
•Application of differential equation,
•Nature and methods of solution,
•Initial-value problems,
•Boundary-value problems,
•Existence of solutions,
Differential Equations
Definition: An equation involving derivatives one or more dependent variable
. With respect to one or more independent variables is called a
. differential equation.
Example: For examples of differential equations we list the following
𝑑𝑦
𝑑𝑥
+y=0
𝜕𝑦
𝜕𝑥
+
𝜕𝑦
𝜕𝑢
+
𝜕𝑦
𝜕𝑣
=0
In the above example it is clear that the various variables and derivatives
involved in a differential equation can occur in a variety of ways.
Order of Differential Equation
The order of the differential equation is order of the highest
derivative in the differential equation.
Differential Equation ORDER
32  x
dx
dy
0932
2
 y
dx
dy
dx
yd
36
4
3
3






 y
dx
dy
dx
yd
1
2
3
Degree of Differential Equation
Differential Equation Degree
032
2
 ay
dx
dy
dx
yd
36
4
3
3






 y
dx
dy
dx
yd
03
53
2
2












dx
dy
dx
yd
1
1
3
The degree of a differential equation is power of the highest order derivative
term in the differential equation.
Classification
Differential equations
Ordinary
DE
Partial
DE
First order
DE
Higher order
DE
Linear
DE
Non-
Linear
DE
Non-
Linear
DE
Linear
DE
First
order
DE
Higher
order
DE
Linear
DE
Non-
Linear
DE
Linear
DE
Non-
Linear
DE
Non-
Linear
DE
Linear
DE
Ordinary DE:
An equation involving ordinary derivatives of one or more dependent variables
. with respect to one or single independent variable is called ordinary differential
. equation.
Examples:
1.
𝑑𝑦
𝑑𝑥
+y=0
2.
d2
y
dx2 +9
𝑑𝑦
𝑑𝑥
+5y=0
3.
𝑑4
𝑥
𝑑𝑡4+9
𝑑2
𝑥
𝑑𝑡2+3x=0
y is dependent variable and x is independent variable,
Independent variables
Dependent variable
Dependent variables
Independent variable
Partial DE:
An equation involving partial derivatives of one or more dependent variable with .
respect to one or more independent variables is called partial differential equation.
Examples:
𝜕𝑦
𝜕𝑥
+
𝜕𝑦
𝜕𝑢
+
𝜕𝑦
𝜕𝑣
=0
𝜕2
𝑢
𝜕𝑥2 +
𝜕2
𝑢
𝜕𝑦2 +
𝜕2
𝑢
𝜕𝑧2 =0
Hence the first equation y is a dependent variable and x,u,v are independent variablesand the
second equation u is a dependent variable and x,y,z are independent variables.
Dependent variable
Independent variable
Dependent variable
Independent variable
First order DE:
An equqtion which involves first order differential coefficient is . .
called first order differential equation.
Examples:
𝒅𝒚
𝒅𝒙
+y=0
𝝏𝒚
𝝏𝒙
+
𝜕𝑦
𝜕𝑢
+
𝜕𝑦
𝜕𝑣
=0
Hence the first and second equation having one order diffrerential coefficient so
both are called first order differential equation.
First order ordinary differential
equation
First order partial differential
Equation
Higher order DE:
An equation which involves higher order differential coefficient
. is called higher order differential equation.
Examples:
1.
d2
y
dx2 +9
𝑑𝑦
𝑑𝑥
+5y=0
2.
𝑑4
𝑥
𝑑𝑡4+9
𝑑2
𝑥
𝑑𝑡2+3x=0
3.
𝜕2
𝑢
𝜕𝑥2 +
𝜕2
𝑢
𝜕𝑦2 +
𝜕2
𝑢
𝜕𝑧2
Here equation (1) is second order ordinary DE, equation(2) is fourth order
ordinary DE and equation(3) is second order partial differential equations.
Higher order differential
equation(Ordinary)
Higher order differential
equation(Partial)
Linear Differential Equation
A differential equation is linear, if
1. dependent variable and its derivatives are of degree one,
2. coefficients of a term does not depend upon dependent
variable.
Example: 36
4
3
3






 y
dx
dy
dx
yd
is non - linear because in 2nd term is not of
degree one.
.0932
2
 y
dx
dy
dx
yd
Example:
is linear.
1
.
2
.
Example:
3
2
2
2
x
dx
dy
y
dx
yd
x 
is non - linear because in 2nd term coefficient depends on y.
3.
Example:
is non - linear because siny is a transcendental function.
y
dx
dy
sin4.
introduction to differential equations
1. Determining the motion of a projectile , rocket, satellite or planet.
2. Determining the charge or current in a electric circuit.
3. Conduction of heat in a rod or in a slab.
4. Determining the vibrations of a wire or membrane.
5. The rate of growth of a population.
6. The rate of the decomposition of a radio-active substance.
7. The study of the reactions of chemicals.
8. Determination of curves that have certain geometrical properties.
Application of Differential Equation
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
introduction to differential equations
Differential Equation Chapter 1 35
Solution of Differential Equation
There are two types of solution of differential equations
1. General solution
2. Particular solution
General Solution:
Examples:
y=ax+bx2 is the solution of differential equation x2 𝑑2
𝑦
𝑑𝑥2- 2x
𝑑𝑦
𝑑𝑥
+ 2y =0, where a
and b are two arbitrary constants.
y=3x+c , is solution of the 1st order
differential equation
𝑑𝑦
𝑑𝑥
= 3 , c is arbitrary constant.
As is solution of the differential equation for every
value of c, hence it is known as general solution.
Particular solution:
Examples:
y=5x2+3x+2 is a particular solution where A=5,B=3 and C=2
Or
y=x2+2x+3 (A=1,B=2,C=3) is the particular solution
A particular solution of differential equation is obtain from the primitive by
assigning definite values to the arbitrary constants.
A relation between the variables which involves n essential arbitrary constants,as
y=x2+cx or y=Ax2+Bx is called a primitive
INITIAL VALUE PROBLEM
• In many physical problems we need to find the particular
solution that satisfies a condition of the form y(x0)=y0.
This is called an initial condition, and the problem of
finding a solution of the differential equation that
satisfies the initial condition is called an initial-value
problem.
• Example (cont.): Find a solution to y2 = x2 + C satisfying the
initial condition y(0) = 2.
22 = 02 + C
C = 4
y2 = x2 + 4
Examples:
𝑑2 𝑦
𝑑𝑥2
+ y = 0
y(1) = 3
y’(1) = -4
This problem consists in finding a solution of the differential equation
𝑑2 𝑦
𝑑𝑥2
+ y = 0
Which assumes the value 3 at x=1 and whose first derivative assumes the value -4 at x=1.Both
of this conditions relate to one x values, namely , x=1.Thus this is an initial-value problem.
Definition:
A boundary value problem is a system of ordinary DE with solution and derivatives
values specified at more than one point most commonly , the solution and derivatives are
specified at just two points (the boundaries)defining two points boundary value i.e the
conditions relate to two different x values, the problem is called a two point boundary value
problem(0r simply a boundary-value problem).
Example:
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
+ 𝐲 = 𝟎
y(0)=1 , y(П/2)=5
The solution must assume the value 1 at x=0 and the value at x= П/2.That is the conditions
relate to the two different x values,0 and П/2.This is a (two point) boundary-value problem.
BOUNDARY VALUE PROBLEM
Existence of solution:
We were able to find a solution of the initial – value problem under
consideration , but do all initial-value and boundary-value problems have
solutions ? We have already observed that this question is in the negative
form.. That is no.
For we have pointed out that the boundary problem
𝑑2 𝑦
𝑑𝑥2 + y = 0 ; y(0) = 1 , y(𝜋) = 5
Mentioned at the ends of solving this problem it observed that it has no
solution ! Thus arises the question of existence of solutions….
The question is – Is there really exist the solutions of the problems ?
This is generally known as EXISTENCE OF SOLUTION
The eternal existential question
If we are given any old initial-value problem
• Does there have to be a solution?
• If so, could there be more than one solution?
(Think of questions like “does 2x5 - 10x + 5 = 0 have a
solution? if so, how many?” We can show that there
is a solution between x=-1 and x=1, but we can’t factor
the polynomial to find it, and we don’t know how
many there are.)

dy
dt
 f (t,y), y(t0 )  y0
The existence theorem
The existence theorem (p. 66) basically says that if f(t, y) is
continuous “near” (t0, y0), then the differential equation
has a solution “near” time t0.
Most of the functions we’ll see in this class are continuous (at least,
most of the time!).

dy
dt
 f (t,y), y(t0 )  y0
Formal statement of the existence theorem
• Check out the theorem on p. 66.
• The statement “there exists an  > 0” means that
there is some positive value the variable  can take
on so that the statement becomes true.
• The theorem does not tell us how large that value
is.
Example
Uniqueness of solutions
OK, so in most reasonable situations, at least one
solution to an IVP will exist.
Did I say at least one????
Does this mean there can be more than one????
YES.
If the function f(t, y) and its partial derivative
are continuous at (t0, y0), a solution exists and is
unique near (t0, y0). Otherwise, there might be
more than one solution!
Try dy/dt = 3y2/3, y(0)=0, to see how this can look.

f y
1 sur 46

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introduction to differential equations

  • 1. EMDADUL HAQUE MILON ehmilon24171@gmail.com mdehmilon24171@gmail.com DEPERTMENT OF STATISTICS, UNIVERSITY OF RAJSHAHI.. RAJSHAHI, BANGLADESH 6205&6206
  • 2. The members of Group-A • Kaderi kibria Farjana Yasmin moni Md. Shadikug jaman Mst. Afia ibnat • Mst. Sumona sumi Minhazul islam Md. Ayub ali nayan Md. Dinar
  • 4. •Topics: •Basic concept of differential equation, •Classification, •Application of differential equation, •Nature and methods of solution, •Initial-value problems, •Boundary-value problems, •Existence of solutions,
  • 5. Differential Equations Definition: An equation involving derivatives one or more dependent variable . With respect to one or more independent variables is called a . differential equation. Example: For examples of differential equations we list the following 𝑑𝑦 𝑑𝑥 +y=0 𝜕𝑦 𝜕𝑥 + 𝜕𝑦 𝜕𝑢 + 𝜕𝑦 𝜕𝑣 =0 In the above example it is clear that the various variables and derivatives involved in a differential equation can occur in a variety of ways.
  • 6. Order of Differential Equation The order of the differential equation is order of the highest derivative in the differential equation. Differential Equation ORDER 32  x dx dy 0932 2  y dx dy dx yd 36 4 3 3        y dx dy dx yd 1 2 3
  • 7. Degree of Differential Equation Differential Equation Degree 032 2  ay dx dy dx yd 36 4 3 3        y dx dy dx yd 03 53 2 2             dx dy dx yd 1 1 3 The degree of a differential equation is power of the highest order derivative term in the differential equation.
  • 8. Classification Differential equations Ordinary DE Partial DE First order DE Higher order DE Linear DE Non- Linear DE Non- Linear DE Linear DE First order DE Higher order DE Linear DE Non- Linear DE Linear DE Non- Linear DE Non- Linear DE Linear DE
  • 9. Ordinary DE: An equation involving ordinary derivatives of one or more dependent variables . with respect to one or single independent variable is called ordinary differential . equation. Examples: 1. 𝑑𝑦 𝑑𝑥 +y=0 2. d2 y dx2 +9 𝑑𝑦 𝑑𝑥 +5y=0 3. 𝑑4 𝑥 𝑑𝑡4+9 𝑑2 𝑥 𝑑𝑡2+3x=0 y is dependent variable and x is independent variable, Independent variables Dependent variable Dependent variables Independent variable
  • 10. Partial DE: An equation involving partial derivatives of one or more dependent variable with . respect to one or more independent variables is called partial differential equation. Examples: 𝜕𝑦 𝜕𝑥 + 𝜕𝑦 𝜕𝑢 + 𝜕𝑦 𝜕𝑣 =0 𝜕2 𝑢 𝜕𝑥2 + 𝜕2 𝑢 𝜕𝑦2 + 𝜕2 𝑢 𝜕𝑧2 =0 Hence the first equation y is a dependent variable and x,u,v are independent variablesand the second equation u is a dependent variable and x,y,z are independent variables. Dependent variable Independent variable Dependent variable Independent variable
  • 11. First order DE: An equqtion which involves first order differential coefficient is . . called first order differential equation. Examples: 𝒅𝒚 𝒅𝒙 +y=0 𝝏𝒚 𝝏𝒙 + 𝜕𝑦 𝜕𝑢 + 𝜕𝑦 𝜕𝑣 =0 Hence the first and second equation having one order diffrerential coefficient so both are called first order differential equation. First order ordinary differential equation First order partial differential Equation
  • 12. Higher order DE: An equation which involves higher order differential coefficient . is called higher order differential equation. Examples: 1. d2 y dx2 +9 𝑑𝑦 𝑑𝑥 +5y=0 2. 𝑑4 𝑥 𝑑𝑡4+9 𝑑2 𝑥 𝑑𝑡2+3x=0 3. 𝜕2 𝑢 𝜕𝑥2 + 𝜕2 𝑢 𝜕𝑦2 + 𝜕2 𝑢 𝜕𝑧2 Here equation (1) is second order ordinary DE, equation(2) is fourth order ordinary DE and equation(3) is second order partial differential equations. Higher order differential equation(Ordinary) Higher order differential equation(Partial)
  • 13. Linear Differential Equation A differential equation is linear, if 1. dependent variable and its derivatives are of degree one, 2. coefficients of a term does not depend upon dependent variable. Example: 36 4 3 3        y dx dy dx yd is non - linear because in 2nd term is not of degree one. .0932 2  y dx dy dx yd Example: is linear. 1 . 2 .
  • 14. Example: 3 2 2 2 x dx dy y dx yd x  is non - linear because in 2nd term coefficient depends on y. 3. Example: is non - linear because siny is a transcendental function. y dx dy sin4.
  • 16. 1. Determining the motion of a projectile , rocket, satellite or planet. 2. Determining the charge or current in a electric circuit. 3. Conduction of heat in a rod or in a slab. 4. Determining the vibrations of a wire or membrane. 5. The rate of growth of a population. 6. The rate of the decomposition of a radio-active substance. 7. The study of the reactions of chemicals. 8. Determination of curves that have certain geometrical properties. Application of Differential Equation
  • 35. Differential Equation Chapter 1 35 Solution of Differential Equation
  • 36. There are two types of solution of differential equations 1. General solution 2. Particular solution General Solution: Examples: y=ax+bx2 is the solution of differential equation x2 𝑑2 𝑦 𝑑𝑥2- 2x 𝑑𝑦 𝑑𝑥 + 2y =0, where a and b are two arbitrary constants. y=3x+c , is solution of the 1st order differential equation 𝑑𝑦 𝑑𝑥 = 3 , c is arbitrary constant. As is solution of the differential equation for every value of c, hence it is known as general solution.
  • 37. Particular solution: Examples: y=5x2+3x+2 is a particular solution where A=5,B=3 and C=2 Or y=x2+2x+3 (A=1,B=2,C=3) is the particular solution A particular solution of differential equation is obtain from the primitive by assigning definite values to the arbitrary constants. A relation between the variables which involves n essential arbitrary constants,as y=x2+cx or y=Ax2+Bx is called a primitive
  • 38. INITIAL VALUE PROBLEM • In many physical problems we need to find the particular solution that satisfies a condition of the form y(x0)=y0. This is called an initial condition, and the problem of finding a solution of the differential equation that satisfies the initial condition is called an initial-value problem. • Example (cont.): Find a solution to y2 = x2 + C satisfying the initial condition y(0) = 2. 22 = 02 + C C = 4 y2 = x2 + 4
  • 39. Examples: 𝑑2 𝑦 𝑑𝑥2 + y = 0 y(1) = 3 y’(1) = -4 This problem consists in finding a solution of the differential equation 𝑑2 𝑦 𝑑𝑥2 + y = 0 Which assumes the value 3 at x=1 and whose first derivative assumes the value -4 at x=1.Both of this conditions relate to one x values, namely , x=1.Thus this is an initial-value problem.
  • 40. Definition: A boundary value problem is a system of ordinary DE with solution and derivatives values specified at more than one point most commonly , the solution and derivatives are specified at just two points (the boundaries)defining two points boundary value i.e the conditions relate to two different x values, the problem is called a two point boundary value problem(0r simply a boundary-value problem). Example: 𝒅 𝟐 𝒚 𝒅𝒙 𝟐 + 𝐲 = 𝟎 y(0)=1 , y(П/2)=5 The solution must assume the value 1 at x=0 and the value at x= П/2.That is the conditions relate to the two different x values,0 and П/2.This is a (two point) boundary-value problem. BOUNDARY VALUE PROBLEM
  • 41. Existence of solution: We were able to find a solution of the initial – value problem under consideration , but do all initial-value and boundary-value problems have solutions ? We have already observed that this question is in the negative form.. That is no. For we have pointed out that the boundary problem 𝑑2 𝑦 𝑑𝑥2 + y = 0 ; y(0) = 1 , y(𝜋) = 5 Mentioned at the ends of solving this problem it observed that it has no solution ! Thus arises the question of existence of solutions…. The question is – Is there really exist the solutions of the problems ? This is generally known as EXISTENCE OF SOLUTION
  • 42. The eternal existential question If we are given any old initial-value problem • Does there have to be a solution? • If so, could there be more than one solution? (Think of questions like “does 2x5 - 10x + 5 = 0 have a solution? if so, how many?” We can show that there is a solution between x=-1 and x=1, but we can’t factor the polynomial to find it, and we don’t know how many there are.)  dy dt  f (t,y), y(t0 )  y0
  • 43. The existence theorem The existence theorem (p. 66) basically says that if f(t, y) is continuous “near” (t0, y0), then the differential equation has a solution “near” time t0. Most of the functions we’ll see in this class are continuous (at least, most of the time!).  dy dt  f (t,y), y(t0 )  y0
  • 44. Formal statement of the existence theorem • Check out the theorem on p. 66. • The statement “there exists an  > 0” means that there is some positive value the variable  can take on so that the statement becomes true. • The theorem does not tell us how large that value is.
  • 46. Uniqueness of solutions OK, so in most reasonable situations, at least one solution to an IVP will exist. Did I say at least one???? Does this mean there can be more than one???? YES. If the function f(t, y) and its partial derivative are continuous at (t0, y0), a solution exists and is unique near (t0, y0). Otherwise, there might be more than one solution! Try dy/dt = 3y2/3, y(0)=0, to see how this can look.  f y